Fermion Masses and Unification

Steve King

University of Southampton

Lecture III Family Symmetry and Unification

1.Introduction to family symmetry2.Froggatt-Nielsen mechanism3.Gauged U(1) family symmetry and its shortcomings4.Gauged SO(3) family symmetry and vacuum alignment 5.A4 and vacuum alignment6. A4 Pati-Salam Theory

Appendix A. A4

Appendix B. Finite Groups

3 3

3 2 2

3 2

0

1

uY

• Universal form for mass matrices, with Georgi-Jarlskog factors

• Texture zero in 11 position

Recall Symmetric Yukawa textures

3 3

3 2 2

3 2

0

1

dY

3 3

3 2 2

3 2

0

3 3

3 1

eY

0.05, 0.15

1( ) , ( ) 3

3s d

GUT GUTe

m mM M

m m

1. Introduction to Family Symmetry

To account for the fermion mass hierarchies we introduce a spontaneously broken family symmetry

It must be spontaneously broken since we do not observe massless gauge bosons which mediate family transitions

The Higgs which break family symmetry are called flavons

The flavon VEVs introduce an expansion parameter = < >/M where M is a high energy mass scale

The idea is to use the expansion parameter to derive fermion textures by the Froggatt-Nielsen mechanism (see later)

In SM the largest family symmetry possible is the symmetry of the kinetic terms

36

1

, , , , , , (3)c c c ci i

i

D Q L U D E N U

In SO(10) , = 16, so the family largest symmetry is U(3)

Candidate continuous symmetries are U(1), SU(2), SU(3) or SO(3) etc

If these are gauged and broken at high energies then no direct low energy signatures

U(1)

SU(2)

SU(3) SO(3)

S(3)Nothing

(3) (3)L RO O

(3) (3)L RS S27

GFamily

4 12A

Simplest example is U(1) family symmetry spontaneously broken by a flavon vev

For D-flatness we use a pair of flavons with opposite U(1) charges

0 ( ) ( )Q Q

Example: U(1) charges as Q (3 )=0, Q (2 )=1, Q (1 )=3, Q(H)=0, Q( )=-1,Q()=1

Then at tree level the only allowed Yukawa coupling is H 3 3 !

0 0 0

0 0 0

0 0 1

Y

The other Yukawa couplings are generated from higher order operators which respect U(1) family symmetry due to flavon insertions:

2 3 4 6

2 3 2 2 1 3 1 2 1 1H H H H HM M M M M

M

When the flavon gets its VEV it generates small effective Yukawa couplings in terms

of the expansion parameter

6 4 3

4 2

3 1

Y

1 0 1 0 0

2.Froggatt-Nielsen Mechanism

What is the origin of the higher order operators?

To answer this Froggat and Nielsen took their inspiration from the see-saw mechanism

2

R

L L

H

M

2 3HM

Where are heavy fermion messengers c.f. heavy RH neutrinos

L LR R

M

H H

RM

2M

H

3

M

There may be Higgs messengers or fermion messengers

2

M

0H

30

1

0

2 3

1

0H

1H1H HM

Fermion messengers may be SU(2)L doublets or singlets

2QQ

M

0H

3cU0

Q

1

0Q 2Q

cU

M

0H

3cU1

cU

1

1cU

3. Gauged U(1) Family SymmetryProblem: anomaly cancellation of SU(3)C

2U(1), SU(2)L2U(1) and U(1)Y

2U(1) anomalies implies that U(1) is linear combination of Y and B-L (only anomaly free U(1)’s available) but these symmetries are family independent

Solution: use Green-Schwartz anomaly cancellation mechanism by which anomalies cancel if they appear in the ratio:

Suppose we restrict the sums of charges to satisfy

Then A1, A2, A3 anomalies are cancelled a’ la GS for any values of x,y,z,u,v

But we still need to satisfy the A1’=0 anomaly cancellation condition.

The simplest example is for u=0 and v=0 which is automatic in SU(5)GUT

since10=(Q,Uc,Ec) and 5*=(L,Dc) qi=ui=ei and di=li so only two independent ei, li.

In this case it turns out that A1’=0 so all anomalies are cancelled.

Assuming for a large top Yukawa we then have:

SO(10) further implies qi=ui=ei=di=li

F=(Q,L) and Fc=(Uc,Dc,Ec,Nc)

In this case it turns out that A1’=0. PS implies x+u=y and x=x+2u=y+v.

So all anomalies are cancelled with u=v=0, x=y. Also h=(hu, hd)

The only anomaly cancellation constraint on the charges is x=y which implies

Note that Y is invariant under the transformations

This means that in practice it is trivial to satisfy

A Problem with U(1) Models is that it is impossible to obtain

3 3

3 2 2

3 2

0

1

Y

For example consider Pati-Salam where there are effectively no constraints on the charges from anomaly cancellation

There is no choice of li and ei that can give the desired texture6 4 3

4 2

3 1

Y

e.g. previous example l1=e1=3, l2=e2=1, l3=e3=hf=0 gave:

Shortcomings of U(1) Family Symmetry

The desired texture can be achieved with non-Abelian family symmetry. Another motivation for non-Abelian family symmetry comes from neutrino physics.

columns

SFKSequential dominance can account for large neutrino mixing

T T T

LL

AA BB CCm

X Y Z See-saw

Sequential dominance Dominant

m3

Subdominant m2

Decoupled m1

Diagonal RH nu basis

Tri-bimaximal

†LV

Constrained SD

Large lepton mixing motivates non-Abelian family symmetry

1

2 2

3 3

0

LR

B

Y A B

A B

Need

Suitable non-Abelian family symmetries must span all three families e.g.

2$ 3 symmetry (from maximal atmospheric mixing)

1$ 2 $ 3 symmetry (from tri-maximal solar mixing)

SFK, Ross; Velasco-Sevilla; Varzelias

SFK, Malinsky

with CSD

27

4

(3)

(3)

SU

SO A

4. Gauged SO(3) family symmetry

Antusch, SFK 04

Left handed quarks and leptons are triplets under SO(3) family symmetryRight handed quarks and leptons are singlets under SO(3) family symmetry

123

a

b

c

23

0

e

f

3

0

0

h

Real vacuum alignment

(a,b,c,e,f,h real)Barbieri, Hall, Kane,

Ross

To break the family symmetry introduce three flavons 3, 23, 123

But this is not sufficient to account

for tri-bimaximal neutrino mixing

2

2

2 3

1

1

0

0

0 i

iLR

i

i

i i

ee

fe

Y

he

ae

be

ce

123

a

b

c

23

0

e

f

3

0

0

h

123. RF h 2

123. RF h 13. RF h

If each flavon is associated with a particular right-handed neutrino

then the following Yukawa matrix results

1 2 323 123 3

1 1 1i i ii R i R i RHL HL HL

M M M

2

2

1

3

2

1

0

0

0 i

Li

iR

i

i i

ve

v

ve

ve

ve

Y

e Ve

123

v

v

v

23

0

v

v

3

0

0

V

123. RF h 2

123. RF h 33. RF h

1

M

For tri-bimaximal neutrino mixing we need

The motivation for 123 is to give the second column required by tri-bimaximal neutrino mixing

How do we achieve such a vacuum alignment of the flavon vevs?

Vacuum Alignment in SO(3)

First set up an orthonormal basis:

1

23

FA=0 flatness <1>=1

FB=0 flatness <2>=2

FC=0 flatness <3>=3

FD=0 flatness 1 .2 =0FE=0 flatness 1 .3 =0FF=0 flatness 2 .3 =0

SFK ‘05

1

1

0

0

2

0

1

0

3

0

0

1

1

23

Then align 23 and 123 relative to 1 , 2 , 3 using additional terms:

FR=0 <23> gets vevs in the (2,3) directions FT=0 <123> gets vevs in the (1,2,3) directions

(vevs of equal magnitude are required to minimize soft mass terms)

23

0

1

1

123

1

1

1

Finally 23 is orthogonal to 123 due to 123

. 23

=0

1

3

223

123

De M.Varzielas, SFK, Ross

We can replace SO(3) by a discrete A4 subgroup:

4 (3)A SO

A4 is similar to the semi-direct product

Same invariants as A4

2=12+2

2+32 ,

3 =1 2 3

The main advantage of using discrete family symmetry groups is that vacuum alignment is simplified…

5. A4 and Vacuum Alignment

The Diamond (A4) Crystal Structure

0,1,1

1,1,1

Radiative Vacuum AlignmentVarzielas, SFK, Ross, Malinsky

0

(s)top loops drive negative

A nice feature of MSSM is radiative EWSB Ibanez-Ross

ZM GUTM

2 2

uHm

2

uHm

H H3LQ

Rt

Similar mechanism can be used to drive flavon vevs using D-terms

Leads to desired vacuum alignment with discrete family symmetry A4

negative

for negative

for positive

23 (0,1, 1)T v

3123

for positive 123

Symmetry group of the tetrahedron

Discrete set of possible vacua

Ma; Altarelli, Feruglio; Varzeilas, Ross, SFK, Malinsky

4A

Comparison of SO(3) and A4

(3,4,2,1)i

L

L

u

e

uF

u

dd d

(1,4,1,2)i

i iR

R

uuF

u

ed d d

SFK, Malinsky

4 (4) (2) (2)PS L RA SU SU SU

6. A4 Pati-Salam Theory

Dirac Operators:

Further Dirac Operators required for quarks:

Dirac Operators: Dirac Neutrino matrix:

. .

. .

Majorana Operators

223 2

2123

0 0

0 0

0 0 1RR

HM

M

•CSD in neutrino sector due to vacuum alignment of flavons

• m3 » m2 » 1/ and m1» 1 is much smaller since ¿ 1

•See-saw mechanism naturally gives m2» m3 since the cancel

Dirac Neutrino matrix:

Majorana Neutrino matrix:

The Messenger Sector

Majorana:

Dirac:

Including details of the messenger sector:

Messenger masses:

Appendix A. A4SFK, Malinsky hep-ph/0610250

Appendix B. Finite Groups Ma 0705.0327